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        Questions tagged [arithmetic-geometry]

        Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

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        52 views

        Heights for rational points via Neron models

        I only just started reading about heights in arithmetic geometry, so forgive me if this question is naive. Suppose $E$ is an elliptic curve over a number field $K$ with ring of integers $R$ and let $\...
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        4answers
        1k views

        Connection Between Knot Theory and Number Theory

        Is there any connection between knot theory and number theory in any aspects? Does anybody know any book that is about knot theory and number theory?
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        1answer
        220 views

        Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$

        My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular: Given a projective variety $V$ defined over $L$ algebraically closed, of ...
        2
        votes
        0answers
        96 views

        Extending section of étale morphism of adic spaces

        This question is related to Lifting points via étale morphism of adic spaces. Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...
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        1answer
        177 views

        On a refinement of Mordell's conjecture for curves

        Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \...
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        votes
        2answers
        300 views

        Down to earth, intuition behind a Anabelian group [closed]

        An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center. I would like to know ...
        6
        votes
        1answer
        200 views

        The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

        Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$. I know that to construct the Jacobian variety associated to $C$, one ...
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        0answers
        67 views

        Igusa curve at infinite level

        In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....
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        votes
        1answer
        238 views

        Smooth proper variety over a number field with prescribed bad reductions

        Given a number field $K$, and a finite set $S$ containing finite places of $K$. When can we find a smooth proper geomerically connected variety $X$ over $K$ such that $X$ has good reduction outside $S$...
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        votes
        2answers
        172 views

        What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?

        Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ? For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
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        votes
        0answers
        95 views

        On the existence of nice hypercovers

        Let $\mathcal{C}$ be a site and $X$ a sheaf of sets on $\mathcal{C}$. Then there exists a hypercover $K_{\bullet}$ of $X$ such that $K_n$ is a coproduct of representable presheaves on $\mathcal{C}$ ...
        3
        votes
        1answer
        195 views

        Nearby cycles and extension by zero

        Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$. Call $i_s ...
        13
        votes
        1answer
        371 views

        Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

        $\newcommand{\Spec}{\operatorname{Spec}}$ Cross-post from Math.SE, hopefully people more knowledgeable in the field will see the question here on MO. It is a well-known fact that a smooth projective ...
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        votes
        0answers
        120 views

        Berthelot’s comparison theorem and functoriality

        Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$. Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
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        0answers
        177 views

        p-adic Hodge theory for singular projective varieties

        In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic ...

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        山西福彩快乐十分钟
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